Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

11. Marriage, Divorce, Children 469
and described in Figure 11.2, which is drawn for the case in which the
transfer from the father, s, exceeds the efficient level of expenditure on the
child good, c ∗ . For levels of a close to yh, the mother spends all her time
on the child and the implied expenditure on the child good c exceeds c ∗ .
In this case, both spouses want to reduce c and increase a, although the
child may be hurt from such a substitution. As a is raised sufficiently so
that c reaches c ∗ the newly formed couple enters the region of conflict.
Any further increase in a and the associated decrease in c, benefits the new
husband but reduces the mother’s (and the child’s) utility. Initially, the
mother continues to spends all her time on the child but when a reaches
yh+s−ĉ, shestartstoworkparttimeandcontinuestodosountila reaches
yh + wm + s − ĉ. Inthissegment,theParetofrontierislinear because the
child good is held at a fixed level, c =ĉ, and any increase in a is achieved by
an increase in hm which raises the father’s utility by wmdhm and reduces
the mother’s utility by ((1 + α)wm − γ)dhm. At high levels of a, exceeding
yh +wm +s−ĉ, the mother works full time in the market and as a rises, the
amount of child good is reduced until it reaches zero and the new husband
obtains all the household resources, yh + wm + s.
To proceed with the analysis, one must determine how the conflict between
the spouses is resolved and which particular point on the Pareto
frontier is selected. For simplicity, we assume that the new husband obtains
all the surplus from remarriage, so that the point on the Pareto frontier
is selected so as to make the mother indifferent between remarriage and
remaining single. This allows us to illustrate the general equilibrium issues
in a relatively simple manner. The reader may interpret the model as a
worst case scenario from the point of view of the mother and child. 11 The
efficient level of adult consumption is then defined as the solution of the
following maximization program
a(s, yh) = max a
a,hm,c
subject to
(11.14)
a + c = wmhm + yh + s,
(1 + α)a + γ(1 − hm)+g(c) ≥ um(s),
0 ≤ hm ≤ 1.
For remarriage to take place, it must be the case that the solution of this
11 We could use instead a symmetric Nash-Bargaining solution to determine the bargaining
outcome (see Chiappori and Weiss, 2007). The Nash axioms imply that the
bargaining outcome must maximize the product of the gains from remarriage, relative
to remaining single, of the two partners. This model yields similar qualitative results,
because the mother is assumed to have lower income and therefore her options outside
marriage are worse than those of men. The magnitudes of the welfare loss of the child
and mother would, of course, be smaller if the mother gets a larger share of the gains
from marriage.